# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Spherical Trig**

**From:**Bill B

**Date:**2005 Apr 5, 20:59 -0500

Thank you NS and Dan, One of those queries that raises the question as to what I was thinking, or was I thinking at all? ;-) If, for example, I had two points on the equator, one at 90d W, another at 90d E, and a third at the pole; I would have 90d each for the equator angles, and another 180d for the LHA/t angle. Hence 360d total or 2 pi radians. If I had LHA approaching 360d, I would have 359.xxx plus 2X 90.yyy, so close to 540d or 3 pi radians. As a follow up question: Dan stated, "The sum of the angles of a spherical triangle is between pi and 3 pi radians (180 deg and 540 deg)." Focusing on the words "is between" does that literally mean the lower limit can approach 180d and the upper limit approach 540d, or can the sums actual equal either 180d or 540d? Bill > On Apr 5, 2005, at 4:08 PM, n s gurnell wrote: > >> Re Bill's question about the angles in a spherical triangle. >> Sixty years ago the exams called "Principles for Second Mates" used to >> have a >> question:- "What is Spherical Excess?" I've forgotten the exact answer >> but >> someone might know. Old Timer > > The sum of the angles of a spherical triangle is between pi and 3 pi > radians (180 deg and 540 deg). The amount by which it exceeds 180 deg > is called the spherical excess and is denoted E. The difference between > 2 pi radians (360 deg) and the sum of the side arc lengths a, b, and c > is called the spherical defect and is denoted D. > > Girard's formula: > > Spherical Excess = E = A + B + C - pi > where A,B, and C are angles measured in radians > Surface Area = E * R^2 > where R is the radius of the sphere > > Dan